The relationships between the number of different spacers in an arrangement of square tiles and the dimensions of the tiles in the same arrangement.

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Introduction

We have been given the task of investigating the relationships between the number of different spacers in an arrangement of square tiles and the dimensions of the tiles in the same arrangement. I will begin my investigation by researching square arrangements of tiles, and then move onto rectangular arrangements. I will then investigate triangle arrangements.

Stage 1 – Square arrangements

The spacers that will be used in this investigation are –

+ Spacer        T Spacer        L Spacer

  I began by drawing 5 different arrangements of tiles, beginning with a 1x1 arrangement, and finishing with a 5x5 arrangement. I drew these on a separate piece of graph paper. (See sheet S1).

  The results gathered from these sketches are shown here:

  The rule for the number of squares is xy. The pattern for the L spacers appears to be a rule, as L spacers only occur on the corners of the arrangements, and being a quadrilateral arrangement there will always be 4 corners. Therefore the rule for the L shaped spacers will be n = 4. For the + spacers the rule will be n = (n-1)  . For the T Spacers the rule will be n = 4n – 4. I will now test these rules.

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  I have now drawn the 10x10 arrangement and have written the actual properties as read from my sketch in this table:

  From the results gathered by actually drawing the arrangement I have deduced that the rules that I obtained from my original set of results are in fact correct for the square arrangements that I have drawn so far, and they appear to be able to hold true when tested with larger arrangements.  I have therefore decided to move onto the next stage of my investigation.

Stage 2 – Rectangular arrangements

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